There are two sides to this answer. One is that A Brief History of Time was first published in 1988, and so it pre-dates the discovery of dark energy by several years. (The question explicitly says that it's not about dark energy, but mentioning it is unavoidable, since it's the reason why most cosmologists no longer predict a big crunch.) At the time Hawking was writing, the data was ambiguous about whether there was enough mass in the Universe to cause the expansion to eventually reverse, resulting in a big crunch. In that case the separation of distant space-time regions would not have been permanent - the separated regions would eventually have become accessible again as the universe shrank in the distant future.
The second side is, does this mean that what Hawking writes about the arrow of time is now out-dated? I would say no: there is still an important scientific question of whether the entropic arrow of time necessarily the same as the cosmological-expansion arrow of time. Even though we no longer think the cosmological arrow will be reversed in the future, we can still use a big crunch scenario as a "thought experiment" to reason about whether the second law would also have to be reversed. When read in this way, Hawking's argument about the relationship between the two arrows is still addressing a relevant question.